First Module Cohomology Group of Induced Semigroup Algebras

Let S be a discrete semigroup and T be a left multiplier on S. A new product on S defined by T creates a new induced semigroup ST. In this paper, we show that if T is bijective, then the first module cohomology groups H1 l1(E)(l1(S), l infinity(S)) and H1l1(ET )(l1(ST ), l infinity(ST )) are equal, where E and ET are sets of idempotent elements in S and ST, respectively. Which in particular means that l1(S) is weak l1(E)-module amenable if and only if l1(ST) is weak l1(ET )-module amenable. Finally, by giving an example, we show that the bijectivity of T, is necessary.